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Examples of nonlinear continuous tensor products of measure spaces and non-Fock factorizations. (English) Zbl 0912.46022

A number of examples for random objects that are nonlinear generalizations of random processes with independent values are constructed in this extensive paper. These objects are a continuous tensor product of measure spaces (measure space factorization) which is not isomorphic to the classical continuous tensor product, and a continuous product of Hilbert spaces (Hilbert factorization). There are several connections of this topic with problems in the probability theory (filtrations), in the representation theory of algebras and groups and in the theory of invariant measures on spaces of solutions for nonlinear hyperbolic equations.
The paper is subdivided into five sections: 1. Continuous tensor products of Hilbert spaces, 2. Measures on flabby sheaves and nets of Borel spaces, 3. Inverse limit constructions and criteria of nonlinearity and continuity of factorizations, 4. Nonlinearizable factorizations over zero-dimensional space, 5. Nonlinearizable factorization over one-dimensional space.
Reviewer: H.Junek (Potsdam)

MSC:

46B28 Spaces of operators; tensor products; approximation properties
46M05 Tensor products in functional analysis
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
47A67 Representation theory of linear operators
60H30 Applications of stochastic analysis (to PDEs, etc.)
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